Time-evolving matrix product operators for off-diagonal system-bath coupling

This paper extends the time-evolving matrix product operator (TEMPO) method within the process tensor framework to solve bosonic quantum impurity problems with off-diagonal system-bath coupling, providing a unified and generic approach that reveals the limitations of secular approximations in sub-ohmic baths and suggests future applications in fermionic generalizations and dynamical mean field theory.

The Gist

The Big Picture: The "Noisy Room" Problem

Imagine you are trying to have a quiet conversation with a friend (the System) in a very loud, chaotic room full of people shouting and moving around (the Bath or Environment).

In quantum physics, we want to know exactly how your friend's conversation changes because of the noise. Usually, we assume the noise is simple: maybe everyone is just whispering the same thing, or the noise is just a steady hum. This is like the "standard" models scientists have used for decades.

However, in the real world, the noise is often messy and complicated. Sometimes the noise interacts with your friend in weird, non-linear ways—like if the crowd starts reacting to your friend's words and then shouting back in a complex pattern. This is called off-diagonal coupling.

Until now, the best computer tools scientists had to simulate this were like trying to solve a puzzle with a hammer: they worked great for simple puzzles but broke down when the puzzle got complex.

The New Tool: "TEMPO" Gets an Upgrade

The authors of this paper have upgraded a famous computer method called TEMPO (Time-Evolving Matrix Product Operator).

Think of the old TEMPO method as a super-efficient librarian.

• The Old Librarian: Could only organize books that were stacked neatly in rows (simple, diagonal noise). If you tried to give them a pile of books thrown in a chaotic heap (complex, off-diagonal noise), they would get confused and the library would collapse.
• The New Librarian (This Paper): Has been given a new set of tools. They can now organize the chaotic heap of books perfectly. They can handle the "messy" interactions where the system and the environment are tangled up in complex ways.

How Does It Work? The "Shadow Puppet" Analogy

To understand how they did it, imagine you are in a dark room with a light source and a wall.

• The System: You (the puppet master).
• The Bath: The light source and the air currents.
• The Shadow: The shadow on the wall (what we can actually see/measure).

Usually, to understand the shadow, you have to track every single air molecule and every photon of light. That's impossible; there are too many of them.

The Process Tensor (the core idea behind TEMPO) is like a magic camera that takes a photo of the shadow without needing to track the air or the light. It captures the "history" of how the shadow moved.

• The Old Way: The camera could only take photos if the light was shining straight down. If the light was coming from the side or bouncing off mirrors (off-diagonal coupling), the camera couldn't focus.
• The New Way: The authors figured out how to build a camera lens that works even when the light is bouncing around wildly. They realized that the "shadow" (the system's history) can be described as a Matrix Product Operator (MPO).

What is an MPO?
Imagine a long chain of people holding hands.

• In the old method, each person just held a simple note (a number).
• In the new method, each person holds a whole instruction manual (a matrix).
  This allows the chain to carry much more complex information about how the noise is affecting the system, without needing to simulate every single noisy particle in the universe.

The "Jaynes-Cummings" Experiment: A Surprise Discovery

To test their new camera, the authors simulated a specific scenario: a tiny spinning top (a Spin) interacting with a sub-ohmic bath (a specific type of noisy environment).

They compared two models:

1. The Standard Model: Assumes the noise interacts in a simple, predictable way (like a gentle breeze).
2. The "JC" Model (Their New Focus): Assumes the noise interacts in a complex, "off-diagonal" way (like a gust of wind that pushes the top sideways and spins it).

The Shocking Result:
They found that for a long time, scientists have been using a shortcut called the "Secular Approximation." This is like assuming, "Hey, the wind is weak, so we can just ignore the weird sideways pushes and only look at the gentle breeze."

The authors proved that this shortcut is dangerous. Even when the noise is very weak, if the environment has a specific "structure" (like their sub-ohmic bath), the "weird sideways pushes" actually change the outcome completely. The standard model predicted the top would spin one way, but their new, more accurate method showed it spinning a totally different way.

The Analogy:
It's like driving a car.

• Standard Model: You assume the road is flat and straight. You drive at a steady speed.
• New Model: You realize the road has hidden bumps and curves. Even if the bumps are small, if you ignore them, you might crash. The "Secular Approximation" is ignoring the bumps, and this paper says, "Don't ignore the bumps, even if they look small!"

Why Should You Care?

1. Better Quantum Computers: Quantum computers are very sensitive to noise. If we want to build them, we need to understand exactly how noise messes them up. This new method gives us a much clearer picture of that mess.
2. Universal Tool: This isn't just for one specific problem. The authors built a "universal adapter." It works for almost any situation where a quantum system talks to a noisy environment, as long as the noise doesn't talk to itself.
3. Future Proofing: They even hinted that this method could be adapted for fermions (a different type of quantum particle, like electrons), which is a huge deal for simulating materials and chemistry.

Summary in One Sentence

The authors built a smarter, more flexible computer program that can simulate how quantum systems behave in messy, complex environments, proving that the old "simplifying shortcuts" scientists used for years often lead to the wrong answers.

Technical Summary

1. Problem Statement

The paper addresses the challenge of simulating the real-time dynamics of Quantum Impurity Problems (QIPs) where a quantum system is coupled to a non-interacting bosonic bath via off-diagonal coupling.

• Context: Traditional methods like the Time-Evolving Matrix Product Operator (TEMPO) have been highly successful for systems with diagonal system-bath coupling (where the system operator coupled to the bath is Hermitian and commutes with itself at different times).
• Limitation: Many physically relevant models, such as the Jaynes-Cummings (JC) model or the Rabi model, involve off-diagonal coupling (non-Hermitian system operators, e.g., $\hat{\sigma}_+$ and $\hat{\sigma}_-$). In these cases, the system operators do not commute, and the standard TEMPO formulation (which relies on interpreting the Feynman-Vernon influence functional as a classical partition function) fails or requires perturbative approximations that lose generality.
• Goal: To develop a non-perturbative, numerically exact method that extends TEMPO to handle general off-diagonal system-bath couplings without relying on the secular approximation or specific details of the coupling Hamiltonian.

2. Methodology

The authors propose a generalized TEMPO framework rooted in the Process Tensor (PT) formalism and the Feynman-Vernon Influence Functional (IF).

A. Theoretical Framework

• Process Tensor (PT): The dynamics of the system are described by a PT, which maps a sequence of quantum operations to the final reduced density matrix. The PT is represented as a Matrix Product Operator (MPO).
• Feynman-Vernon IF: For bosonic baths, the influence of the environment is encoded in the IF.
  • Diagonal Case: The IF can be interpreted as the partition function of a classical Hamiltonian, naturally represented as a Matrix Product State (MPS).
  • Off-Diagonal Case: The authors demonstrate that for off-diagonal coupling, the IF corresponds to the thermal state of an effective quantum many-body Hamiltonian ($\hat{H}_{\text{eff}}$). Consequently, the IF must be represented as an MPO rather than an MPS.

B. Algorithm: Extended TEMPO

The core algorithm involves constructing the MPO representation of the IF ($I[\hat{A}^\dagger, \hat{A}]$) using the following steps:

1. Discretization: The continuous-time IF is discretized on the Keldysh contour using the QuAPI (Quasi-Adiabatic Propagator Path-Integral) scheme. This results in a discrete expression resembling a thermal state $e^{-\hat{H}_{\text{eff}}}$ with an inverse temperature of 1.
2. MPO Construction via XTRG: Instead of standard time-evolution, the authors use the Exponential Tensor Renormalization Group (XTRG) algorithm.
   • They construct $e^{-\hat{H}_{\text{eff}}/2^m}$ as an MPO.
   • They iteratively multiply these MPOs ($m$ times) to obtain the full $e^{-\hat{H}_{\text{eff}}}$.
   • Compression: To manage computational cost, intermediate MPOs (with bond dimension $\chi^2$) are compressed back to bond dimension $\chi$ using singular value truncation. The authors utilize an iterative minimization scheme for MPO-MPO multiplication to reduce the scaling from $O(N\chi^6d^3)$ to $O(N\chi^4d^3)$.
3. System Dynamics: The system Hamiltonian $\hat{H}_S$ is absorbed into the MPO-IF via local operations on the physical indices, yielding the final Process Tensor.

C. Key Distinctions from Previous Work

• Generality: Unlike previous extensions that relied on perturbative treatments of the system path integral, this method relies solely on the Trotter decomposition of the IF. It is blind to the specific details of the system-bath coupling, making it applicable to phenomenological models and Bosonic Dynamical Mean Field Theory (BDMFT).
• MPO vs. MPS: The fundamental shift is recognizing that off-diagonal coupling requires an MPO representation (quantum thermal state) rather than an MPS (classical partition function).

3. Key Contributions

1. Generalization of TEMPO: The first non-perturbative extension of TEMPO to handle general off-diagonal system-bath couplings (non-Hermitian system operators).
2. Unified Framework: The method naturally reduces to standard TEMPO and its diagonal extensions when the coupling becomes diagonal, providing a unified theoretical understanding of all TEMPO variants.
3. Fermionic Generalization: The paper outlines a direct path to generalize this method to fermionic impurity problems (e.g., Anderson impurity models), suggesting a "Fermionic TEMPO" that has not yet been fully explored.
4. Algorithmic Efficiency: The use of XTRG and iterative MPO multiplication allows for efficient construction of the influence functional with controlled bond dimensions.

4. Numerical Results

The authors validated the method on three distinct cases:

• Case 1: Jaynes-Cummings (JC) Model (Single Mode Bath)

  • Setup: A spin coupled to a single bosonic mode.
  • Result: The extended TEMPO results matched Exact Diagonalization (ED) perfectly for both weak ($\lambda^2=0.1$) and strong ($\lambda^2=0.5$) coupling.
  • Observation: The method achieved high accuracy with a relatively small bond dimension ($\chi=30$), demonstrating that the PT information can be significantly compressed even for oscillating hybridization functions.

• Case 2: Non-interacting Bosonic Mode (Continuous Sub-Ohmic Bath)

  • Setup: A free boson system coupled to a sub-Ohmic bath ($J(\omega) \propto \omega^s$).
  • Result: The method converged rapidly with respect to time step ($\delta t$) and bond dimension ($\chi$). Results for average occupation and Green's functions matched ED benchmarks.
  • Efficiency: Accurate results were obtained with $\chi=30$, whereas standard ED requires discretizing the bath into many modes, introducing truncation errors.

• Case 3: JC Spin-Boson Model vs. Standard Spin-Boson Model

  • Setup: A spin coupled to a sub-Ohmic bath via JC-type coupling (off-diagonal) vs. Rabi-type coupling (diagonal).
  • Key Finding: The authors compared the JC model (often used as an approximation for the Rabi model via the secular approximation) against the full Rabi model.
  • Result: Even in the weak coupling regime ($\alpha=0.01$), the dynamics of the JC model deviated significantly from the standard spin-boson model at long times ($t > 3$).
  • Implication: The secular approximation fails in the presence of a structural (sub-Ohmic) bath, even when the coupling is weak. The physics of the two models diverges due to non-Markovian effects and the conservation of excitation numbers in the JC model.

5. Significance and Impact

• Breaking the Secular Approximation: The work provides rigorous evidence that the secular approximation (rotating wave approximation) is insufficient for describing quantum impurity dynamics in structured environments, even at weak coupling. This challenges common practices in quantum optics and condensed matter physics.
• Unified Solver for BDMFT: The method serves as a robust, non-perturbative impurity solver for Bosonic Dynamical Mean Field Theory (BDMFT), capable of handling off-diagonal hybridization functions without perturbative assumptions.
• Future Directions: By establishing the MPO-IF framework for off-diagonal coupling, the paper opens the door for simulating complex fermionic systems (like the Anderson impurity model) using similar tensor network techniques, potentially revolutionizing the study of non-equilibrium quantum many-body systems.

In summary, this paper provides a mathematically rigorous and computationally efficient extension of the TEMPO method, bridging the gap between diagonal and off-diagonal system-bath couplings and revealing critical limitations of standard approximations in non-Markovian quantum dynamics.